A concept of fuzzy transition probabilities

Starting from fuzzy real numbers with an arbitrary lattice of belief and following the extension principle, we develop concepts of fuzzy probabilities, transition probabilities and random variables and of their combinations, and show that these concepts are consistent. We derive some results on fuzzy real numbers, on the expectation of fuzzy random variables and on fuzzy stochastic processes. To sketch the range of applications of fuzzy stochastics, we give two examples that show how real-world problems may be modeled by means of fuzzy probabilities and that give small numerical examples. Moreover, we give a brief outlook for a possible expansion of our theory to fuzzy Markovian decision processes by means of a partial order on the set of all fuzzy real numbers.     

Mixed aleatory and epistemic uncertainty quantification using fuzzy set theory

This paper proposes algorithms to construct fuzzy probabilities to represent or model the mixed aleatory and epistemic uncertainty in a limited-size ensemble. Specifically, we discuss the possible requirements for the fuzzy probabilities in order to model the mixed types of uncertainty, and propose algorithms to construct fuzzy probabilities for both independent and dependent datasets. The effectiveness of the proposed algorithms is demonstrated using one-dimensional and high-dimensional examples. After that, we apply the proposed uncertainty representation technique to isocontour extraction, and demonstrate its applicability using examples with both structured and unstructured meshes.     

Transition probabilities for Markov chains having fuzzy states

We consider a Markov Chain in which the states are fuzzy subsets defined on some finite state space. Building on the relationship between set-valued Markov chains to the Dempster-Shafer combination rule, we construct a procedure for finding transition probabilities from one fuzzy state to another. This construction involves Dempster-Shafer type mass functions having fuzzy focal elements. It also involves a measure of the degree to which two fuzzy sets are equal. We also show how to find approximate transition probabilities from a fuzzy state to a crisp state in the original state space     

A Fuzzy Logic Approach for Determining Expected Values: A Project Management Application

This paper presents a methodology rooted in the general concepts of fuzzy logic theory with specific emphasis on belief functions and extension principles, and fuzzy probability distributions with fuzzy expectation based on fuzzy probability measures. This approach offers a useful alternative to the traditional approach in the estimation of probabilities in the absence of information about relative frequencies. An algorithm called BIPFET is developed and its application is demonstrated by utilizing data from a real-life research and development project.     

The solution of fuzzy linear systems by non-linear programming: a financial application

Fuzzy linear systems of equations play a major role in various financial applications. In this paper we analyse a particular fuzzy linear system: the derivation of the risk neutral probabilities in a fuzzy binary tree. This system has previously been investigated and different solutions to different forms of the same system have been proposed.     

Statistical maps and generalized random walks

We continue our study of statistical maps (equivalently, fuzzy random variables in the sense of Gudder and Bugajski). In the realm of fuzzy probability theory, statistical maps describe the transportation of probability measures on one measurable space into probability measures on another measurable space. We show that for discrete probability spaces each statistical map can be represented via a special matrix the rows of which are probability functions related to conditional probabilities and the columns are related to fuzzy n-partitions of the domain. Discrete statistical maps sending a probability measure p to a probability measure q can be represented via conditional distributions and correspond to joint probabilities on the product. The composition of statistical maps provide a tool to describe and to study generalized random walks and Markov chains.     

Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment

As a generalized fuzzy number, the hesitant fuzzy element (HFE) has been receiving increased attention and has recently become a popular topic. However, we find that the occurring probabilities of the possible values in the HFE are equal, which is obviously impractical. Consequently, in this paper, we propose a hesitant fuzzy number with probabilities, called the hesitant probabilistic fuzzy number, and construct its score function, deviation function, comparison laws, and its basic operations. It is well known that in the context of a group of decision makers (DMs), one of the basic approaches to built consensus is to aggregate individual evaluations or individual priorities. Thus, to use the hesitant fuzzy numbers for consensus building with a group of DMs, we further propose a method called maximizing score deviation method to obtain the DMs’ weights under the HPFE environment, based on which two extended and four new ordered weighted operators are provided to fuse the HPFE information and build the consensus of the DMs. We also analyze the differences among these ordered weighted operators and provide their application scopes. Finally, a practical case is provided to demonstrate consensus building with a group of DMs under the HPFE environment using the proposed approaches.     

Fuzzy decision trees

We propose here to extend the decision trees method to the case when the involved data (probabilities, cost, profits, losses) appear as words belonging to the common language whose semantic representations are fuzzy sets. First we discuss the reasons why such an extension is to be aimed at. Then in the fuzzy case we carry out a reformalization of the basic concepts of probability and utility theory. Finally we show how these reformalized concepts can be applied to fuzzy decision trees.     

模糊数学理论在水库长期径流预报中的应用

本文运用模糊数学的理论方法对汛期径流预报进行了探讨 ,即首先对汛期来水的理论实测资料 ,运用最大树法和软化分法进行了模糊聚类分析 ,得到汛期来水的多个预报模式 ;进而利用Gamma分布和模糊概率的方法 ,预报出汛期的来水流量及其实现的概率     

Using fuzzy set theory in a scheduling problem: A case study

This paper deals with a real scheduling problem where it seems interesting to use fuzzy sets The question of knowing how and when it is possible to use fuzzy sets (rather than probabilities for instance) is discussed in great detail for the studied case. Fuzzy concepts are shown to be very useful and easy to work with in this decision-aid problem.     

模糊环境下美式看跌期权的定价研究

应用模糊集理论将无风险利率和波动率进行模糊化,以梯形模糊数替代精确值,将美式期权的定价模型扩展到美式期权模糊定价模型.得到了模糊风险中性概率表达式,并在此概率测度下推导出多期二叉树模糊定价模型,以及二叉树上各节点以梯形模糊数表示的模糊期权价值,以数值模拟演示了美式看跌期权的模糊定价过程.最后分析了不同风险偏好投资者在不确定环境下的套利决策行为,结果表明风险偏好大的投资者具有较高的置信水平、较小的主观模糊期权价格以及较大的无风险套利区间.     

信度理论的扩张与一个审定损害的专家系统

通过使用集合的基本概率定义出上限、下限概率,形成信度;通过隶属函数定义模糊集合的包含度、相交度,使信度理论在模糊集合得以扩张,得到了利用不确定性及模糊性的一个合理的推理方法。在此基础上,采AND/OR/COMB树推理开发了一个审定损害的专家系统。     

Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility

This paper considers Stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random two-level linear programming problems. On the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal, together with the introduction of probability maximization criterion in stochastic programming, we propose new two-level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values. Through the proposed models, it is shown that the original two-level linear programming problems with fuzzy random variables can be transformed into deterministic two-level linear fractional programming problems. For the transformed problems, extended concepts of Stackelberg solutions are defined and computational methods are also presented. A numerical example is provided to illustrate the proposed methods.     

Stability analysis of fuzzy Markovian jumping Cohen–Grossberg BAM neural networks with mixed time-varying delays

In this paper, we investigate the robust stability of uncertain fuzzy Markovian jumping Cohen–Grossberg BAM neural networks with discrete and distributed time-varying delays. A new delay-dependent stability condition is derived under uncertain switching probabilities by Takagi–Sugeno fuzzy model. Based on the linear matrix inequality (LMI) technique, upper bounds for the discrete and distributed delays are calculated using the LMI toolbox in MATLAB. Numerical examples are provided to illustrate the effectiveness of the proposed method.     

模糊集IF-THEN规则的蚁群空间聚类分析

将蚂蚁的拾起和放下对象的行为表示为模糊集.通过模糊集的IF-THEN规则计算蚂蚁执行任务的激励和反应阈值,得到蚂蚁拾起或放下项目的概率,对蚂蚁的行为做出决策,实现对空间数据的聚类.以矿山实际测量数据为空间数据源,采用基本的蚁群聚类算法和模糊蚁群空间聚类算法分别对其进行聚类.通过对这两种算法的实验结果进行分析比较,证明改进后的算法提高了聚类效果.     

Robust stability and stabilization analysis for discrete-time randomly switched fuzzy systems with known sojourn probabilities

The stability and stabilization analysis problem is considered in this paper for a class of discrete-time switched fuzzy systems with known sojourn probabilities. By using Lyapunov functional, new delay-dependent sufficient conditions are derived to ensure the stability of the system. Convex combination technique is incorporated and the stability criteria are presented in terms of Linear matrix inequalities (LMIs). Furthermore, the developed approach is extended to address the robust stability and stabilization analysis of the delayed discrete-time switched fuzzy systems with randomly occurring uncertainties. Finally numerical examples are exploited to substantiate the theoretical results.     

Some mathematical tools for linguistic probabilities

We propose and develop, in this paper, some concepts and techniques useful for the theory of linguistic probabilisies introduced by L.A. Zadeh. These probabilities are expressed in linguistic rather than numerical terms. The mathematical framework for this study is based upon the possibility theory.We formulate first the problem of optimization under elastic constraints which is not only important for mathematical programming but will be served to justify the extension of possibility measure to linguistic variables. Next, in connection with translation rules in natural languages we study some transformations of fuzzy sets using a relation between random sets and fuzzy sets. Finally, we point out some differences between random variables and fuzzy variables, and present the mathematical notion of possibility, in the setting of set-functions, as a special case of Choquet capacities.     

Nonparametric criteria for supervised classification of fuzzy data

The supervised classification of fuzzy data obtained from a random experiment is discussed. The data generation process is modelled through random fuzzy sets which, from a formal point of view, can be identified with certain function-valued random elements. First, one of the most versatile discriminant approaches in the context of functional data analysis is adapted to the specific case of interest. In this way, discriminant analysis based on nonparametric kernel density estimation is discussed. In general, this criterion is shown not to be optimal and to require large sample sizes. To avoid such inconveniences, a simpler approach which eludes the density estimation by considering conditional probabilities on certain balls is introduced. The approaches are applied to two experiments; one concerning fuzzy perceptions and linguistic labels and another one concerning flood analysis. The methods are tested against linear discriminant analysis and random K-fold cross validation.     

Generalized fuzzy set systems and particularization

It is suggested that there exists many fuzzy set systems, each with its specific pointwise operations for union and intersection. A general law of compound possibilities is valid for all these systems, as well as a general law for representing marginal possibility distributions as unions of fuzzy sets. Max-min fuzzy sets are a special case of a fuzzy set system which uses the pointwise operations of max and min for union and intersection respectively. Probabilistic fuzzy sets are another special case which uses the operations of addition and multiplication. Probably there exists an infinite number of fuzzy set operations and systems. It is shown why the law of idempotency for intersection is not required for such systems. An essential difference between the meaning of the operations of union and intersection in traditional measure theory as compared with their meaning in the theory of possibility is pointed out. The operation of particularization is used to illustrate that the two distinct classical theories of nonfuzzy relations and of probability are merely two aspects of a more generalized theory of fuzzy sets. It is shown that we must distinguish between particularization of conditional fuzzy sets and of joint fuzzy sets. The concept of restriction of nonfuzzy relations is a special case of particularization of both conditional and joint fuzzy sets. The computation of joint probabilities from conditional and marginal ones is a special case of particularization of conditional probabilistic fuzzy sets. The difference between linguistic modifiers of type 1 and particulating modifiers is pointed out, as well as a general difference between nouns and adjectives.     

Optimization under consideration of uncertain input quantities

This contribution presents an approach to account for imprecise data within an optimization task in view of engineering applications. In order to specify imprecise data the concept of imprecise probabilities is utilized, applying the generalized uncertainty model fuzzy randomness. Considering the fact, that the uncertainty affects both the objective function and the constraints, the optimum and the respective design is obtained imprecise. In view of decision making for engineering applications the optimization is converted to account for information reducing methods, e.g. determination of failure probabilities, defuzzification and robustness assessment. The introduced methods and algorithms are focused on a numerical treatment to solve nonlinear industry–sized problems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)